Brazilian Journal of Physics, vol. 34, no. 3A, September, 2004 855
Dirac-Hartree-Bogoliubov Approximation for
Finite Nuclei with Blocking
E. Baldini-Neto, B. V. Carlson,
Departamento de F´ısica, Instituto Tecnol´ogico de Aeron´autica,
Centro T´ecnico Aeroespacial, 12228-900 S˜ao Jos´e dos Campos, S˜ao Paulo, Brazil
and D. Hirata
Department of Physics and Astronomy, The Open University,
Milton Keynes Mk7 6AA, Buckinghamshire, United Kingdom
Received on 03 October, 2003
We have extended the Dirac-Hartree-Bogoliubov (DHB) approximation developed in a previous work [1] with the inclusion of blocking terms in order to study the pairing properties of both even and odd nuclei. We have concentrated our attention in the isotope chains of12−26
O,34−56
Ca,48−78
Ni and100−132
Sn as well as on the α-decay of the new superheavy element277112.
1
Introduction
The development of new facilities with the aim of studying unstable nuclei has enabled experimental measurements of masses, radii and deformations of such systems in a wider region of the nuclear chart. Studies in this ‘terra exotica’ have revealed new features such as neutron halos or skins [2, 3, 4] and brought new perspectives to nuclear physics.
Relativistic many-body theories have been applied to nu-clei and nuclear matter with remarkable success [5, 6, 7]. In a previous work [1], the Dirac-Hartree-Fock-Bogoliubov (DHFB) approximation was developed to describe the ground state wave functions and energies of finite nuclei. It was applied to spin-zero proton-proton and neutron-neutron pairing within the Dirac-Hartree-Bogoliubov (DHB) ap-proximation (the exchange term was neglected) using a zero-range approximation to the pairing tensor. The latter can be justified by demonstrating that the effective length for
spatial variations of the wavefunctions, in the calculations performed, is much larger than the range of the non-locality of the pairing tensor, rendering the spatial variations of the wave function close to negligible within the range of pair-ing non-locality. The Dirac structure of the pairpair-ing field is retained in the DHFB approximation. The resulting pairing field is dominated by a scalar term and the zero component of a vector term, as found in the case for1
S0pairing in
sym-metric nuclear matter.
2
Dirac-Hartree with blocking
The Lagrangian density can be written as:
L=L0+Lint (1)
whereL0is the free Lagrangian density given by
⌋
L0 = Ψ(¯ x)[i/∂−M]Ψ(x) +
1
2∂µσ(x)∂
µσ(x)]−U(σ(x))−1
4FµνF
µν
−1 4ΩµνΩ
µν+1
2m
2
ωωµ(x)ωµ(x)−
1
4Gµν·G
µν+1
2m
2
ρρµ(x)·ρµ(x), (2)
⌈
with
Fµν =∂µAν−∂νAµ,
Ωµν =∂µων−∂νωµ, (3)
Gµν=∂µρν−∂νρµ,
856 E. Baldini-Netoet al.
Lint=gσΨ(¯ x)σ(x)Ψ(x)−gωΨ(¯ x)γµωµ(x)Ψ(x)−
1
2gρΨ(¯ x)γµτ·ρ
µ(x)Ψ(x)
−eΨ(¯ x)(1 +τ3) 2 γµA
µ(x)Ψ(x). (4)
⌈
The DHFB approximation is obtained by characteriz-ing the average effect of the interaction of a nucleon with the other nucleons through an effective single particle La-grangian,Lef f, given in terms of the self-energy, which de-scribes the average interaction of a nucleon with the
sur-round matter and the pairing field∆(and its conjugate∆¯) which describes the creation (annihilation) of a pair during the propagation. It was shown in Ref.[1] that these fields satisfy the following self-consistency equations,
⌋
Σ(x, y) =δ(x−y)
j
Γjα
d3
zDαβj (x−z)
ǫγ<0
¯
Uγ(z)Γjβ(z)Uγ(z)
−
j
Γα(x)Dαβ(x−y)
ǫγ<0
Uγ(x) ¯Uγ(y)Γjβ, (5)
and
∆(x, y) =−
j
Γjα(x)Dαβj (x−y)
ǫγ<0
Uγ(xV¯γ(y)AΓTjβ(y)A†, (6)
where the sum runs over the negative frequency solutions,ǫγ<0. The generalized baryon (quasi-particle) propagator is
S(˜x,y˜;ω) =
G(x, y;ω) F(x, y;ω) ˜
F(x, y;ω) G˜(x, y;ω)
,
and can be written as
S(˜x,˜y;ω) =
α
Uα(x)
Vα(x)
1
ω−ǫα+iη
¯
Uα(y) V¯α(y)
+
β
Uβ(x)
Vβ(x)
1
ω+ǫβ−iη
¯
Uβ(y) V¯β(y)
⌈
with the componentsUα,βandVα,βbeing Dirac spinors cor-responding to the normal and time-reversed components of the positive (negative) frequencies,ǫα(ǫβ) respectively, of the Dirac-Gorkov equation whose solutions occur in pairs with real eigenvalues of opposite signs. The pairs of eigen-vectors, for neutrons or protons, are related as.
ǫ=ǫα:
U(y)
V(y)
ǫ=−ǫα:
−γ0BV∗(y)
γ0BU∗(y)
.
whereB =γ5
C.
In order to treat odd systems we have performed Pauli-blocking of selected single-particle states, interchanging one of a pair of eigenvectors by its time-reversed state. This cor-responds to the following transformation
U(y)
V(y)
→
−γ0BV∗(y)
γ0BU∗(y)
, (7)
which modifies the densities that enter in the Hartree terms of the self-energy given in equation (5) and annuls the con-tribution of the blocked states to the pairing field, given in equation (6).
3
Results
As a first test of our approach, we have calculated the mass deffects of the 12−26
O, 34−56
Ca, 48−78
Ni and 100−132
Brazilian Journal of Physics, vol. 34, no. 3A, September, 2004 857
10 12 14 16 18 20 22 24 26 28 30
A -5
0 5 10 15 20 25 30 35 40
∆
(MeV)
Expt DHB+BCS
O Isotopes
Figure 1. DHB+BCS calculations for mass deffects of the12−26 O (Z=8) isotope chain compared with the experimental values of Ref. [8]..
30 35 40 45 50 55 60
A -50
-40 -30 -20 -10 0 10 20
∆
(MeV)
Expt DHB+BCS
Ca Isotopes
Figure 2. DHB+BCS calculations for mass deffects of the34−56 Ca (Z=20) isotope chain compared with the experimental values of Ref. [8].
48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 A
-60 -40 -20 0
∆
(MeV)
Expt DHB+BCS
Ni Isotopes
FIgure 3. DHB+BCS calculations for mass deffects of the48−78Ni
(Z=28) isotope chain compared with the experimental values of Ref. [8].
100 104 108 112 116 120 124 128 132 136 140 A
-100 -90 -80 -70 -60 -50 -40
∆
(MeV)
Expt DHB+BCS
Sn Isotopes
Figure 4. DHB+BCS calculations for mass deffects of the 100−132
Sn (Z=50) isotope chain compared with the experimental values of Ref. [8].
We have also calculated theα-decay chain of of the new superheavy element 277
112 [9] within the blocked+DHB formalism. The results, compared to those of a non-relativistic Hartree-Fock-Bogoliubov approach, which con-siders a Gogny force (HFB+Gogny), and also to the exper-imental data of Ref. [8], are shown in Table 1. As can be seen, the results are very promising.
TABLE 1.α-decay chain of277112calculated in blocked+DHB, HFB+Gogny compared with the experimental data of Ref. [8].
mother→daughter gs→gs gs→gs expt (MeV)
blocked+DHB HFB+Gogny 277
112→273112 12.246 12.323 11.45−11.65 273
110→269Hs 10.187 11.266 9.73−11.08(?) 269
Hs→265Sg 10.099 9.034 9.17−9.23(?) 265
Sg→261Rf 8.462 9.084 8.77 261
Rf→257No 6.711 8.487 8.52 257
No→253Fm 7.068 8.048 8.34−8.45 253
Fm→249Cf 8.308 7.446 7.197 249
Cf→245Cm 6.350 6.350 6.295
4
Conclusions and Perspectives
We have included Pauli-blocking terms in the Dirac-Hartree-Bogoliubov approach previously developed in Ref.
[1]. As a initial test we have calculated the mass deffects of the isotope chains of O, Ca, Ni and Sn and theα-decay chain of the superheavy element277
858 E. Baldini-Netoet al.
in the former case, and with a non-relativistic calculation as well in the latter case. As a next step, after repeating the cal-culations shown in this work within the full DHFB approxi-mation, we intend to apply the formalism along the nuclear chart, giving special attention to light neutron-rich isotopes such as those of Li, Be, B, C and N.
EBN acknowledges the support of Fundac¸˜ao de Am-paro `a Pesquisa do Estado de S˜ao Paulo (FAPESP). BVC acknowledges partial support from FAPESP and from the Conselho Nacional de Pesquisa e Desenvolvimento (CNPq).
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